Abstract
In this article, we study mirror symmetry for pairs of singular Calabi-Yau varieties which are double covers of toric manifolds. Their period integrals can be seen as certain 'fractional' analogues of those of ordinary complete intersections. This new structure can then be used to solve their Riemann-Hilbert problems. The latter can then be used to answer definitively questions about mirror symmetry for this class of Calabi-Yau varieties.